Our goal is to rigorously justify the compressible Euler limit and the acoustic limit from the BFD equation. Then proving that the remainders \(F_{R,\varepsilon }\)  of expansion (REF ) will go to zero as \(\varepsilon \rightarrow 0\)  is the key point of this paper. Once this point is showing mathematically, a class of special solutions of BFD equation (REF )-(REF ) are found according to the same discussions as remarkable work in Caflisch [1]}.
